A Risk-Neutral Parametric Liquidity Model for Derivatives

We develop a parameterised model for liquidity effects
arising from the trading in an asset. Liquidity is defined via a combination
of a trader's individual transaction cost and a price slippage impact, which
is felt by all market participants. The chosen definition allows liquidity
to be observable in a centralised order-book of an asset as is usually
provided in most non-specialist exchanges. The discrete-time version of the
model is based on the CRR binomial tree and in the appropriate
continuous-time limits we derive various nonlinear partial differential
equations. Both versions can be directly applied to the pricing and hedging
of options; the nonlinear nature of liquidity leads to natural bid-ask
spreads that are based on the liquidity of the market for the underlying and
the existence of (super-)replication strategies. We test and calibrate our
model empirically with high-frequency data of German blue chips and discuss
further extensions to the model, including stochastic liquidity.